Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [171,4,Mod(64,171)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(171, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 2]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("171.64");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | |||
Weight: | |||
Character orbit: | 171.f (of order , degree , minimal) |
Newform invariants
comment: select newform
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
Self dual: | no |
Analytic conductor: | |
Analytic rank: | |
Dimension: | |
Relative dimension: | over |
Coefficient field: | |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: |
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Coefficient ring: | |
Coefficient ring index: | |
Twist minimal: | yes |
Sato-Tate group: |
-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Coefficients of the -expansion are expressed in terms of a basis for the coefficient ring described below. We also show the integral -expansion of the trace form.
Basis of coefficient ring in terms of a root of
:
Character values
We give the values of on generators for .
Embeddings
For each embedding of the coefficient field, the values are shown below.
For more information on an embedded modular form you can click on its label.
comment: embeddings in the coefficient field
gp: mfembed(f)
Label | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
64.1 |
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−2.72529 | − | 4.72034i | 0 | −10.8544 | + | 18.8003i | −1.48079 | − | 2.56480i | 0 | 6.79546 | 74.7206 | 0 | −8.07113 | + | 13.9796i | ||||||||||||||||||||||||||||||||||||||||||||||
64.2 | −1.71809 | − | 2.97581i | 0 | −1.90364 | + | 3.29720i | 6.83777 | + | 11.8434i | 0 | −20.1525 | −14.4069 | 0 | 23.4957 | − | 40.6958i | |||||||||||||||||||||||||||||||||||||||||||||||
64.3 | −1.45636 | − | 2.52249i | 0 | −0.241981 | + | 0.419123i | −10.1021 | − | 17.4973i | 0 | −1.64299 | −21.8921 | 0 | −29.4246 | + | 50.9649i | |||||||||||||||||||||||||||||||||||||||||||||||
64.4 | 1.45636 | + | 2.52249i | 0 | −0.241981 | + | 0.419123i | 10.1021 | + | 17.4973i | 0 | −1.64299 | 21.8921 | 0 | −29.4246 | + | 50.9649i | |||||||||||||||||||||||||||||||||||||||||||||||
64.5 | 1.71809 | + | 2.97581i | 0 | −1.90364 | + | 3.29720i | −6.83777 | − | 11.8434i | 0 | −20.1525 | 14.4069 | 0 | 23.4957 | − | 40.6958i | |||||||||||||||||||||||||||||||||||||||||||||||
64.6 | 2.72529 | + | 4.72034i | 0 | −10.8544 | + | 18.8003i | 1.48079 | + | 2.56480i | 0 | 6.79546 | −74.7206 | 0 | −8.07113 | + | 13.9796i | |||||||||||||||||||||||||||||||||||||||||||||||
163.1 | −2.72529 | + | 4.72034i | 0 | −10.8544 | − | 18.8003i | −1.48079 | + | 2.56480i | 0 | 6.79546 | 74.7206 | 0 | −8.07113 | − | 13.9796i | |||||||||||||||||||||||||||||||||||||||||||||||
163.2 | −1.71809 | + | 2.97581i | 0 | −1.90364 | − | 3.29720i | 6.83777 | − | 11.8434i | 0 | −20.1525 | −14.4069 | 0 | 23.4957 | + | 40.6958i | |||||||||||||||||||||||||||||||||||||||||||||||
163.3 | −1.45636 | + | 2.52249i | 0 | −0.241981 | − | 0.419123i | −10.1021 | + | 17.4973i | 0 | −1.64299 | −21.8921 | 0 | −29.4246 | − | 50.9649i | |||||||||||||||||||||||||||||||||||||||||||||||
163.4 | 1.45636 | − | 2.52249i | 0 | −0.241981 | − | 0.419123i | 10.1021 | − | 17.4973i | 0 | −1.64299 | 21.8921 | 0 | −29.4246 | − | 50.9649i | |||||||||||||||||||||||||||||||||||||||||||||||
163.5 | 1.71809 | − | 2.97581i | 0 | −1.90364 | − | 3.29720i | −6.83777 | + | 11.8434i | 0 | −20.1525 | 14.4069 | 0 | 23.4957 | + | 40.6958i | |||||||||||||||||||||||||||||||||||||||||||||||
163.6 | 2.72529 | − | 4.72034i | 0 | −10.8544 | − | 18.8003i | 1.48079 | − | 2.56480i | 0 | 6.79546 | −74.7206 | 0 | −8.07113 | − | 13.9796i | |||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
19.c | even | 3 | 1 | inner |
57.h | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 171.4.f.h | ✓ | 12 |
3.b | odd | 2 | 1 | inner | 171.4.f.h | ✓ | 12 |
19.c | even | 3 | 1 | inner | 171.4.f.h | ✓ | 12 |
57.h | odd | 6 | 1 | inner | 171.4.f.h | ✓ | 12 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
171.4.f.h | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
171.4.f.h | ✓ | 12 | 3.b | odd | 2 | 1 | inner |
171.4.f.h | ✓ | 12 | 19.c | even | 3 | 1 | inner |
171.4.f.h | ✓ | 12 | 57.h | odd | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on :
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